# Fiches de cours

## Coxeter groups

Lachowska Anna

English

#### Summary

Study groups generated by reflections

#### Content

- Orthogonal transformations in a real Euclidean space

- Groups generated by reflections. Coxeter groups, root systems. Crystallographic groups. Fundamental regions for Coxeter groups.

- Coxeter graphs. Classification of finite root systems. Classification of finite crystallographic Coxeter groups. Order and structure of irreducible Coxeter groups. Generators and relations of Coxeter groups.

- Affine Coxeter groups. Classification.

- Applications and connections with other fields.

#### Keywords

Orthogonal transformations, reflection, regular polytop, root system, simple root, positive root, Coxeter group, Coxeter graph, crystallographic group, fundamental region, symmetric group, simply laced root system, Bruhat order, the longest element of a Coxeter group, Coxeter element, Coxeter plane, Coxeter number, McKay correspondence

#### Learning Prerequisites

##### Required courses

Linear algebra I-II, Group theory

##### Recommended courses

Linear algebra I-II, Geometry I-Ii, Group theory, Lie algebras, Linear representations of finite groups

#### Learning Outcomes

By the end of the course, the student must be able to:
• Apply concepts and ideas of the course
• Reason rigorously using the notions of the course
• Choose an appropriate method to solve problems
• Identify the concepts relevant to each problem
• Apply known methods to solve problems similar to the examples shown in the course and in the problem sets
• Solve new problems using the ideas of the course
• Implement appropriate methods to identify and study the groups generated by reflections

#### Teaching methods

Lectures and exercise sessions

#### Assessment methods

Written exam

Dans le cas de l¿art. 3 al. 5 du Règlement de section, l¿enseignant décide de la forme de l¿examen qu¿il communique aux étudiants concernés.

#### Supervision

 Office hours No Assistants Yes Forum No

#### Resources

##### Bibliography

1. J. Humphreys, Reflection Groups and Coxeter Groups, Cambridge University Press, 1990.

2. C.T. Benson, L.C. Grove, Finite Reflection Groups. Second Edition, Springer, 2010.

3. A. Bjorner, F. Brenti, Combinatorics of Coxeter Groups. Springer, 2005.

### Dans les plans d'études

• Mathématiques, 2018-2019, Bachelor semestre 6
• Semestre
Printemps
• Forme de l'examen
Ecrit
• Crédits
5
• Matière examinée
Coxeter groups
• Cours
2 Heure(s) hebdo x 14 semaines
• Exercices
2 Heure(s) hebdo x 14 semaines

LuMaMeJeVe
8-9
9-10
10-11
11-12
12-13
13-14MAA330
14-15
15-16MAA330
16-17
17-18
18-19
19-20
20-21
21-22
Cours
Exercice, TP
Projet, autre

### légende

• Semestre d'automne
• Session d'hiver
• Semestre de printemps
• Session d'été
• Cours en français
• Cours en anglais
• Cours en allemand